4 research outputs found
Fractional variational principle of Herglotz for a new class of problems with dependence on the boundaries and a real parameter
The fractional variational problem of Herglotz type for the case where the Lagrangian depends on generalized fractional derivatives, the free endpoints conditions, and a real parameter is studied. This type of problem generalizes several problems recently studied in the literature. Moreover, it allows us to unify conservative and non-conservative dynamical processes in the same model. The dependence of the Lagrangian with respect to the boundaries and a free parameter is effective and transforms the standard Herglotz鈥檚 variational problem into a problem of a different nature.publishe
The Variable-Order Fractional Calculus of Variations
This book intends to deepen the study of the fractional calculus, giving
special emphasis to variable-order operators. It is organized in two parts, as
follows. In the first part, we review the basic concepts of fractional calculus
(Chapter 1) and of the fractional calculus of variations (Chapter 2). In
Chapter 1, we start with a brief overview about fractional calculus and an
introduction to the theory of some special functions in fractional calculus.
Then, we recall several fractional operators (integrals and derivatives)
definitions and some properties of the considered fractional derivatives and
integrals are introduced. In the end of this chapter, we review integration by
parts formulas for different operators. Chapter 2 presents a short introduction
to the classical calculus of variations and review different variational
problems, like the isoperimetric problems or problems with variable endpoints.
In the end of this chapter, we introduce the theory of the fractional calculus
of variations and some fractional variational problems with variable-order. In
the second part, we systematize some new recent results on variable-order
fractional calculus of (Tavares, Almeida and Torres, 2015, 2016, 2017, 2018).
In Chapter 3, considering three types of fractional Caputo derivatives of
variable-order, we present new approximation formulas for those fractional
derivatives and prove upper bound formulas for the errors. In Chapter 4, we
introduce the combined Caputo fractional derivative of variable-order and
corresponding higher-order operators. Some properties are also given. Then, we
prove fractional Euler-Lagrange equations for several types of fractional
problems of the calculus of variations, with or without constraints.Comment: The final authenticated version of this preprint is available online
as a SpringerBrief in Applied Sciences and Technology at
[https://doi.org/10.1007/978-3-319-94006-9]. In this version some typos,
detected by the authors while reading the galley proofs, were corrected,
SpringerBriefs in Applied Sciences and Technology, Springer, Cham, 201
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Application of the generalized Melnikov method to weakly damped parametrically excited cross waves with surface tension
The Wiggins-Holmes extension of the generalized
Melnikov method (GMM) is applied to weakly damped
parametrically excited cross waves with surface tension in
a long rectangular wave channel in order to determine if
these cross waves are chaotic. The Lagrangian density
function for surface waves with surface tension is
simplified by transforming the volume integrals to surface
integrals and by subtracting the zero variation integrals.
The Lagrangian is written in terms of the three generalized
coordinates (or, equivalently the three degrees of freedom)
that are the time-dependent components of the velocity
potential. A generalized dissipation function is assumed to
be proportional to the Stokes material derivative of the
free surface. The generalized momenta are calculated from
the Lagrangian and the Hamiltonian is determined from a
Legendre transformation of the Lagrangian. The first order
ordinary differential equations derived from the
Hamiltonian are usually suitable for the application of the
GMM. However, the cross wave equations of motion must be
transformed in order to obtain a suspended system for the
application of the GMM. Only three canonical
transformations that preserve the dynamics of the cross
wave equations of motion are made because of an extension
of the Herglotz algorithm to nonautonomous systems. This
extension includes two distinct types of the generalized
Herglotz algorithm (GHA). The system of nonlinear
nonautonomous evolution equations determined from
Hamilton's equations of motion of the second kind are
averaged in order to obtain an autonomous system. The
unperturbed system is analyzed to determine hyperbolic
saddle points that are connected by heteroclinic orbits
The perturbed Hamiltonian system that includes surface
tension satisfies the KAM nondegeneracy requirements; and
the Melnikov integral is calculated to demonstrate that the
motion is chaotic. For the perturbed dissipative system
with surface tension, the Melnikov integral is identically
zero implying that a higher dimensional GMM is necessary in
order to demonstrate by the GMM that the motion is chaotic.
However, numerical calculations of the largest Liapunov
characteristic exponent demonstrate that the perturbed
dissipative system with surface tension is also chaotic. A
chaos diagram is computed in order to search for possible
regions of the damping parameter and the Floquet parametric
forcing parameter where chaotic motions may exist